W

The preceding equation of motion is called Newton's law of motion and is governed by the equilibrium of inertia force that is a product of the mass W/g, and acceleration x, and the resisting forces that are a function of the stiffness of the spring.

The principle of virtual work can be used as an alternative to derive Newton's law of motion. Although the method was first developed for static problems, it can readily be applied to dynamic problems by using D'Alembert's principle. The method establishes dynamic equilibrium by including inertial forces in the system.

The principle of virtual work can be stated as follows: For a system in equilibrium, the work done by all the forces during a virtual displacement is equal to zero. Consider

Figure 2.67. Damped oscillator: (a) analytical model; (b) forces in equilibrium.

a damped oscillator subjected to a time-dependent force F{(), as shown in Fig. 2.67. The free-body diagram of the oscillator subjected to various forces is shown in Fig. 2.67b.

Let Sx be the virtual displacement. The total work done by the system is zero and is given by mX SX + cX Sx + kx Sx - F(t) Sx = 0 (2.53)

Since Sx is arbitrarily selected, mx + cx + kx - F(t) = 0 (2.55)

This is the differential equation of motion of the damped oscillator.

The equation of motion for an undamped system can also be obtained from the principle of conservation of energy. It states that if no external forces are acting on the system, and there is no dissipation of energy due to damping, then the total energy of the system must remain constant during motion and consequently, its derivative with respect to time must be equal to zero.s

Consider again the oscillator shown in Fig. 2.67 without the damper. The two energies associated with this system are the kinetic energy of the mass and the potential energy of the spring.

The kinetic energy of the spring

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